# Appleton-Hartree equation

Appleton-Hartree equation

The Appleton-Hartree equation, sometimes also referred to as the Appleton-Lassen equation is a mathematical expression that describes the refractive index for electromagnetic wave propagation in a cold magnetized plasma. The Appleton-Hartree equation was developed independently by several different scientists, including Edward Victor Appleton, Douglas Hartree and K. Lassen.

Equation

Full Equation

The equation is typically given as follows [ Citation | last = Helliwell | first = Robert | date = 2006 | title = Whistlers and Related Ionospheric Phenomena | edition = 2nd | publication-place = Mineola, NY | publisher = Dover | pages = 23-24 ] :

:$n^2 = 1 - frac\left\{X\right\}\left\{1 - iZ - frac\left\{frac\left\{1\right\}\left\{2\right\}Y^2sin^2 heta\right\}\left\{1 - X - iZ\right\} pm frac\left\{1\right\}\left\{1 - X - iZ\right\}left\left(frac\left\{1\right\}\left\{4\right\}Y^4sin^4 heta + Y^2cos^2 hetaleft\left(1 - X - iZ ight\right)^2 ight\right)^\left\{1/2$

Definition of Terms

$n$ = complex refractive index

$i$ = $sqrt\left\{-1\right\}$

$X = frac\left\{omega_0^2\right\}\left\{omega^2\right\}$

$Y = frac\left\{omega_H\right\}\left\{omega\right\}$

$Z = frac\left\{ u\right\}\left\{omega\right\}$

$u$ = electron collision frequency

$omega = 2pi f$

$f$ = wave frequency

$omega_0 = 2pi f_0 = sqrt\left\{frac\left\{Ne^2\right\}\left\{epsilon_0 m$ = electron plasma frequency

$omega_H = 2pi f_H = frac\left\{B_0 |e\left\{m\right\}$ = electron gyro frequency

$epsilon_0$ = permittivity of free space

$mu_0$ = permeability of free space

$B_0$ = ambient magnetic field strength

$e$ = electron charge

$m$ = electron mass

$heta$ = angle between the ambient magnetic field vector and the wave vector

Modes of Propagation

The presence of the $pm$ sign in the Appleton-Hartree equation gives two separate solutions for the refractive index [ Citation | last = Bittencourt | first = J.A. | date = 2004 | title = Fundamentals of Plasma Physics | edition = 3rd | publication-place = New York, NY | publisher = Springer-Verlag | pages = 419-429 ] . For propagation perpendicular to the magnetic field, i.e., $kperp B_0$, the '+' sign represents the "ordinary mode," and the '-' sign represents the "extraordinary mode." For propagation parallel to the magnetic field, i.e., $kparallel B_0$, the '+' sign represents a left-hand circularly polarized mode, and the '-' sign represents a right-hand circularly polarized mode. See the article on electromagnetic electron waves for more detail.

Reduced Forms

Propagation in a Collisionless Plasma

If the wave frequency of interest $omega$ is much smaller than the electron collision frequency $u$, the plasma can be said to be "collisionless." That is, given the condition

$u ll omega$,

we have

$Z = frac\left\{ u\right\}\left\{omega\right\} ll 1$,

so we can neglect the $Z$ terms in the equation. The Appleton-Hartree equation for a cold, collisionless plasma is therefore,

:$n^2 = 1 - frac\left\{X\right\}\left\{1 - frac\left\{frac\left\{1\right\}\left\{2\right\}Y^2sin^2 heta\right\}\left\{1 - X\right\} pm frac\left\{1\right\}\left\{1 - X\right\}left\left(frac\left\{1\right\}\left\{4\right\}Y^4sin^4 heta + Y^2cos^2 hetaleft\left(1 - X ight\right)^2 ight\right)^\left\{1/2$

Quasi-Longitudinal Propagation in a Collisionless Plasma

If we further assume that the wave propagation is primarily in the direction of the magnetic field, i.e., $heta approx 0$, we can neglect the $Y^4sin^4 heta$ term above. Thus, for quasi-longitudinal propagation in a cold, collisionless plasma, the Appleton-Hartree equation becomes,

:$n^2 = 1 - frac\left\{X\right\}\left\{1 - frac\left\{frac\left\{1\right\}\left\{2\right\}Y^2sin^2 heta\right\}\left\{1 - X\right\} pm Ycos heta\right\}$

References

;Citations and notes

ee also

Plasma (physics)

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